Inaba H
Math Biosci. 1989 Oct;96(2):195-219. doi: 10.1016/0025-5564(89)90059-x.
The weak ergodic theorems of mathematical demography state that the age distribution of a closed population is asymptotically independent of the initial distribution. In this paper, we provide a new proof of the weak ergodic theorem of the multistate population model with continuous time. The main tool to attain this purpose is a theory of multiplicative processes, which was mainly developed by Garrett Birkhoff, who showed that ergodic properties generally hold for an appropriate class of multiplicative processes. First, we construct a general theory of multiplicative processes on a Banach lattice. Next, we formulate a dynamical model of a multistate population and show that its evolution operator forms a multiplicative process on the state space of the population. Subsequently, we investigate a sufficient condition that guarantees the weak ergodicity of the multiplicative process. Finally, we prove the weak and strong ergodic theorems for the multistate population and resolve the consistency problem.
数学人口统计学的弱遍历定理表明，封闭人口的年龄分布渐近地独立于初始分布。在本文中，我们给出了具有连续时间的多状态人口模型弱遍历定理的一个新证明。实现这一目标的主要工具是乘法过程理论，该理论主要由加勒特·伯克霍夫发展而来，他表明遍历性质通常适用于一类适当的乘法过程。首先，我们在巴拿赫格上构建乘法过程的一般理论。接下来，我们制定多状态人口的动力学模型，并表明其演化算子在人口的状态空间上形成一个乘法过程。随后，我们研究保证乘法过程弱遍历性的充分条件。最后，我们证明多状态人口的弱遍历定理和强遍历定理，并解决一致性问题。
